Cohen-Macaulay ring
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In mathematics, a Cohen-Macaulay ring is a commutative noetherian local ring with Krull dimension equal to its depth. The depth is always bounded above by the Krull dimension; equality provides some interesting regularity conditions on the ring, enabling some powerful theorems to be proven in this rather general setting.
Examples
- Every regular local ring is Cohen-Macaulay.
- A field is a particular example of a regular local ring, so is Cohen-Macaulay.
- If k is a field, then the formal power series ring in one variable k[[X]] is a regular local ring and so is Cohen-Macaulay, but is not a field.
- Any Gorenstein ring is Cohen-Macaulay.
- Any 0-dimensional ring is Cohen-Macaulay.
- Following the last idea, if k is a field and X is an indeterminate, the ring k[X]/(X2) is a 0-dimensional local ring and so is Cohen-Macaulay, but it is not regular.
- If k is a field, then the formal power series ring k[[t2, t3]], where t is an indeterminate, is an example of a 1-dimensional local ring which is not regular but is Gorenstein, so is Cohen-Macaulay.
- If k is a field, then the formal power series ring k[[t3, t4, t5]], where t is an indeterminate, is an example of a 1-dimensional local ring which is not Gorenstein but is Cohen-Macaulay.
- More generally, any 1-dimensional (Noetherian local) integral domain is Cohen-Macaulay.
The naming here is, in part, for F. S. Macaulay, who worked in elimination theory. The other half is for Irving S. Cohen, one of Zariski's students from his days at Johns Hopkins University.
One meaning of the Cohen-Macaulay condition is seen in coherent duality theory, where it corresponds to the dualizing object, which a priori lies in a derived category, being represented by a single module (coherent sheaf). The finer Gorenstein condition is then expressed by this module being projective (an invertible sheaf). Non-singularity (regularity) is still stronger--it corresponds to the notion of smoothness of a geometric object at a particular point. Thus, in a geometric sense, the notions of Gorenstein and Cohen-Macaulay capture increasingly larger sets of points than the smooth ones, points which are not necessarily smooth but behave in many ways like smooth points anyway.
External link
- MathWorld page (http://mathworld.wolfram.com/Cohen-MacaulayRing.html)